\section{Conclusion}
\begin{itemize}
  \item $\mathcal{L}_\gillies$ and $\mathcal{L}_\Box$ are equivalent (under
    translation)
    \begin{itemize}
      \item $\Box \varphi$ iff $\top \gillies \varphi$
      \item $\varphi \gillies \psi$ iff $\Box(\varphi \to \psi)$ (Gillies'
        Equivalence)
    \end{itemize}
  \item The \emph{unrestricted} Gillies conditional does not have
    reflection in general\ldots
    \begin{itemize}
      \item See proposition 5.5 in \cite{gillies_epistemic_2004}
      \item This provides a challenge for lifting our grammar restrictions
    \end{itemize}
  \item \ldots but it can express Veltman's Might and our $\Box$
    \begin{itemize}
      \item Might $\varphi$ iff $(\varphi \gillies \bot) \gillies \bot$
      \item Hence lifting the grammar restrictions is tantalizing,
        though challenging
    \end{itemize}
  \item Generalizing our results to the other semantic entailment relations
    defined in \cite{van_der_does_updatemight_1997} might be challenging
\end{itemize}

\subsubsection{Mission Accomplished}
  The goal of our research project was to provide a completeness theorem
  for the Gillies conditional, which we have provided.  Moreover, we
  have also illustrated that the Gillies conditional, in the languages
  we have studied, is best understood by understanding the semantics for
  $\Box$ first.
